= Solution
The <divergence theorem> states that, for a sufficiently smooth <vector field> $\mathbf F$ on a volume $\Omega$ with piecewise smooth closed boundary $\partial\Omega$,
$$
\boxed{\oiint_{\partial\Omega}\mathbf F\cdot\hat{\mathbf n}\,dS=\iiint_\Omega\nabla\cdot\mathbf F\,dV}.
$$
Here $\hat{\mathbf n}$ is the outward unit normal. The left-hand side is the total outward <flux>; the right-hand side adds the local <divergence> throughout the volume. The orientation and the requirement to include the complete closed boundary are essential.
Back to article page
